Many-body localized (MBL) states are dynamical phases of isolated, interacting quantum systems in which they do not approach equilibrium from generic initial conditions. MBL states thus defy various expectations from conventional statistical mechanics -- for example, their response to slowly varying perturbations violates linear-response theory. I will present recent results on the response of many-body localized (MBL) systems to time-dependent external perturbations. First, I will discuss the low-frequency behavior of the linear-response optical conductivity and related dynamical response functions. Second, I will explore the regime of validity of linear-response theory, arguing that the conductivity has no well-defined d.c. limit, and discussing the crossovers between the linear-response regime and the nonlinear regime in which saturation effects are dominant. If time permits, I will also discuss the dynamics of localized systems subject to external classical noise.